If x = cos θ + i sin θ, then xⁿ + \(\frac { 1 }{ { x }^{ n } } \) is ______

The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

If ω is the cube root of unity, then the value of (1-ω) (1-ω²) (1-ω⁴) (1-ω⁸) is _________

.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________

If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________

The locus of the point of intersection of perpendicular tangents of the parabola y² = 4ax is

The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is ________

The equation of tangent at (1, 2) to the circle x² + y² = 5 is __________

The area of the circle (x - 2)² + (y - k)² = 25 is _________

\({ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right) \)

The value of sin 2(tan^-1 0.75) is ___________

In an ellipse, the distance between its foci is 6 and its minor axis is 8, then e is ________

If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________

If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________

If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________

If A = [2 0 1] then the rank of AA^T is ______

In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z is non zero then the system is ______

If p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

If ax² + bx + c = 0, a, b, c \(\in\) R has no real zeros, and if a + b + c < 0, then __________

Which of the following is not an elementary transformation?

If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots

The quadratic equation whose roots are ∝ and β is ___________

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________

Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________

If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is

If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I₂ - A = 

If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A = 

If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) = 

If the coordinates at one end of a diameter of the circle x² + y² − 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x² − 9y² = 144 are the roots of x² − (a + b)x − 4 = 0, then the value of (a+b) is

An ellipse has OB as semi minor axes, F and F′ its foci and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

Let C be the circle with centre at(1, 1) and radius = 1. If T is the circle centered at (0, y) passing through the origin and touching the circle C externally, then the radius of T is equal to

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is

If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is

If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to

If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

The principal argument of the complex number \(\frac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

The polynomial x³ + 2x + 3 has

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_o g)(x), then the degree of h is

If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂ + 4z₁z₃ + z₂z₃| = 12, then the value of |z₁+z₂+z₃| is 

If \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is